1. Introduction

Over the few decades, automobile wheel design has progressively evolved from early spoke designs of wood and steel wheels of the horse drawn carriages and bicycle technology, to flat steel discs and in more recent years to the stamped metal configurations and the newer generation of cast and forged aluminium alloy wheels  . Wheel elements, nomenclature, configuration and functions have been described in literature  - . The transition from steel to aluminum alloy wheels, and improved shape design for optimum force flux and stress resistance has been reported to reduce weight up to 50, and 16%, respectively  - . The style, weight, manufacture and performance are the four main technical issues related to the design of new automobile wheels and/or their optimization. Figure 1 shows the critical technical issues in wheel design  - . Advantages of automobile aluminum alloy wheel which include light weight that enhances fuel economy, its noncorrosive characteristics has been widely reported  - .

For modern wheel design the determination of optimum wheel design parameters is a very cumbersome and challenging exercise. Analytical methods are known to be complex and strong theoretical background required, while experimental methods are costly, destructive and time consuming. To this effect, application of numerical

Block Diagram of wheel characteristics

methods such as finite element method (FEM) is currently gaining ground in structural analysis of automobile wheels  - . However, in the application of numerical methods, the formulation of appropriate loading function, which describes the actual radial load distribution, representing the vehicle’s and passengers’ weight, on the numerical model has been a major challenge, hence different approximate loading functions ELF, CLF, polynomial and Hetzian functions  and BLF  - have been developed by different researchers.

In actual wheel, since a radial load is applied to the wheel on the bead seats with tire, the distributed pressure is loaded directly on the bead seat of the model. The pressure is assumed to have a cosine function distribution mode within a central angle of 40˚ in the circumferential direction, Figure 2 (below).

By using the cosine function accordingly, the distributed pressure, Wr, is given as:

The total radial load, W, is evaluated using Equation (2) as follows,

Substituting Equation (2) into Equation (3) results in,

Integrating,

or solving for W0, gives,

where, rb is the bead seat radius and b is the total width of the bead seats.

This function uses the analogy of a half-plane under the action of concentrated force perpendicular to the boun-

dary. For the half plane (Figure 3) the stress function for this problem can be in the form:

Minus sign is chosen because σr obviously will be compressive.

Using Equation (7), we obtain for stresses,

the differential equations and compatibility equation are satisfied identically.

it could be seen that these conditions are satisfied everywhere on line AB, except at point of application of force P.

where, ;substituting expression for σr gives,

Hence, and

substituting (10) into (8(a)) we have,

This is based on round rod in an eye bar under an equilibrium of forces as shown in Figure 4   .

In the Figure 4, r, is the radius of the hole, W is the load imparted, θ, is the angle and qmax is the maximum pointload. The horizontal components of q are balanced. The vertical forces can be related to the external load, W given as:

Defining q = qmaxcosθ and substituting into Equation (12) gives,

Evaluating,

The ELF and CLF have been employed in wheel analysis   , however, in view of the complexity associated with wheel modeling, no one as at yet defined, accurately, the loading function. It is with this in mind that this study takes to undertake a comparison of the various loading arrangement in order to identify the potentially suited function for wheel design and analysis.

2. Material and Method

A selected automobile aluminum alloy wheel configuration (6JX14H2; ET 42) was used in the analysis. The wheel was sectioned and the cut portions taken to the laboratory to determine the mechanical properties with the aid of a universal testing machine and, chemical properties using the spectrophotometric analysis test. Some of the values obtained-yield stress, poison’s ration and density―form part of the input parameters for the analysis. The actual wheel dimensions were obtained using coordinate measuring machine. The 3-D solid model of the wheel was generated, discretised into elements and analysed with commercially available Finite Element software, PTC® (Creo Elements/Pro 5.0). The model consists of 3785 hexahedral elements. The loading condition for the CLF, BLF and ELF were modeled at different angles (30 and 40 deg for CLF and BLF and 90 deg’ for ELF) and inflation pressure of 0.3 and 0.15 MPa, with a radial load of 4750 N. Figure 5 shows the selected wheel and 3-D computer model.

(a) selected aluminium alloy wheel; (b) Computer model of selected wheel; (c) Loaded computer model showing the constraints.
3. Results and Discussion

Relation between Plots of the loading functions and the Sherwood curve on the Von mises Stress at the inboard bead seat at 0.3 MPa inflation pressure and 4750 radial load Relation between Plots of the loading functions and the Sherwood curve on the Von mises Stress at the inboard bead seat at 0.15 MPa inflation pressure and 4750 radial load Mechanical properties of Alloy wheel
Mechanical propertyValue
Young’s Modulus22.29 GPa
Yield Stress222.5 MPa
Poison’s ratio0.42
Ultimate tensile stress69.2 MPa
Percentage elongation2.8 %
Brinell hardness48

MPa at about 4 degree symmetric with the point of contact with the ground (Figure 8).

Relation between Plots of the loading functions and the Sherwood curve on the Von mises Stress at the inboard bead seat at 0 MPa inflation pressure and 4750 radial load Chemical properties of Alloy wheel
ElementPercentage composition (%)
Aluminum (Al) - 87.087.00
Silicon (Si) - 11.1511.150
Cupper (Cu) - 0.4960.496
Magnesium (Mn) - 0.2810.281
Manganese (Mg) - 0.0320.032
Chromium (Cr) - 0.050.050
Zinc (Zn) - 0.2590.259
Titanium (Ti) - 0.0820.082
Iron (Fe) - 0.590.590
Others - 0.020.020

a value of about 0.15 mm at about 90 degree and then dropping to a value of about 0.1 mm. It coincides with the CLF value at about 140 degree and having the characteristics as for the CLF between 160 and 180 degree. The ELF follows the same trend with lower values and with a maximum displacement, at 0 degree, of about 0.20 mm (Figure 9).

The character and shape of the respective curves for CLF, BLF and ELF at 0.15 MPa and 0 MPa inflation are the same as for those of 0.3 MPa inflation pressure. The maximum displacement for ELF, BLF and ELF at 0.15 MPa inflation pressure were about 0.43 mm, 0.375 and 0.175 respectively (Figure 10), while the maximum displacement values, at 0 MPa inflation pressure were about 0.425 mm, 0. 35 and 0.15 mm were obtained respectively for CLF, BLF and ELF at point of contact with the ground (Figure 11). The deformed wheel is shown in Figures 12(a)-(d).

4. Conclusion

Analysis of different loading functions―CLF, BLF and ELF at deferent inflation pressure of 0.3, 0.15 MPa and 0 MPa at specified radial load of 4750N was carried out on a selected aluminium alloy wheel. Von Mises stress was used as a basis for comparison of the different loading functions investigated with the experimental data obtained by Sherwood et al. while the displacement fields (as obtained from the FEM tool) were used as a